The numerical index of the 𝐿_{𝑝} space

Abstract

Abstract. We give a partial answer to the problem of computing the numerical index of L p [0, 1] for 1 < p < ∞.

Numerical radius and a notion of smoothness in the space of bounded linear operators

“…The space of bounded linear operators on X endowed with the numerical radius norm is denoted by (L(X)) w . For a detailed study on numerical range of operators and their possible applications, we refer the readers to [4,5,11,13,14]. Birkhoff-James orthogonality is an important tool in the study of smoothness of elements in a given Banach space.…”

Numerical index of Banach spaces and duality

“…L-and M-spaces have numerical index 1 [6], a property shared by the disk algebra [5, theorem 3•3], and by every Banach space nicely embedded into any C b ( )-space [37, corollary 2•2] (even by every space that is semi-nicely embedded into any C b ( )-space [23, corollary 2]). Very recently, approximations to the computation of the numerical index of the L p (µ)-spaces have been made [8,9], and the exact computation of the numerical indices of the two-dimensional spaces whose unit balls are regular polygons appears in [25].…”

Numerical radius and a notion of smoothness in the space of bounded linear operators

“…This is also true as far as our next result is concerned. It was proved in [4] that numerical radius defines a norm on L(ℓ p ), where 1 ≤ p < ∞. In the next theorem, we prove the same, using support functionals.…”

“…In the next theorem, we prove the same, using support functionals. We believe that the following proof is simpler in comparison to the proof given in [4].…”